Proof Complexity Lower Bounds from Algebraic Circuit Complexity

TL;DR

This paper establishes new lower bounds on algebraic proof systems by translating algebraic circuit complexity results into proof complexity, revealing limitations of certain proof subsystems.

Contribution

It introduces two methods to convert algebraic circuit lower bounds into proof complexity lower bounds for various restricted algebraic proof systems.

Findings

Lower bounds for sparse polynomials, depth-3 powering formulas, and multilinear formulas.

Strict separations between different algebraic proof subsystems.

Short refutations of subset-sum axioms using IPS subsystems.

Abstract

We give upper and lower bounds on the power of subsystems of the Ideal Proof System (IPS), the algebraic proof system recently proposed by Grochow and Pitassi, where the circuits comprising the proof come from various restricted algebraic circuit classes. This mimics an established research direction in the boolean setting for subsystems of Extended Frege proofs, where proof-lines are circuits from restricted boolean circuit classes. Except one, all of the subsystems considered in this paper can simulate the well-studied Nullstellensatz proof system, and prior to this work there were no known lower bounds when measuring proof size by the algebraic complexity of the polynomials (except with respect to degree, or to sparsity). We give two general methods of converting certain algebraic lower bounds into proof complexity ones. Our methods require stronger notions of lower bounds, which lower bound a polynomial as well as an entire family of polynomials it defines. Our techniques are reminiscent of existing methods for converting boolean circuit lower bounds into related proof complexity results, such as feasible interpolation. We obtain the relevant types of lower bounds for a variety of classes (sparse polynomials, depth-3 powering formulas, read-once oblivious algebraic branching programs, and multilinear formulas), and infer the relevant proof complexity results. We complement our lower bounds by giving short refutations of the previously-studied subset-sum axiom using IPS subsystems, allowing us to conclude strict separations between some of these subsystems.